Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi
In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar ${rm O}$-mixed and ${rm H}$-mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of ${rm O}$-mixed and ${rm H}$-mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus.
{"title":"From annular to toroidal pseudo knots","authors":"Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi","doi":"arxiv-2409.03537","DOIUrl":"https://doi.org/arxiv-2409.03537","url":null,"abstract":"In this paper, we extend the theory of planar pseudo knots to the theories of\u0000annular and toroidal pseudo knots. Pseudo knots are defined as equivalence\u0000classes under Reidemeister-like moves of knot diagrams characterized by\u0000crossings with undefined over/under information. In the theories of annular and\u0000toroidal pseudo knots we introduce their respective lifts to the solid and the\u0000thickened torus. Then, we interlink these theories by representing annular and\u0000toroidal pseudo knots as planar ${rm O}$-mixed and ${rm H}$-mixed pseudo\u0000links. We also explore the inclusion relations between planar, annular and\u0000toroidal pseudo knots, as well as of ${rm O}$-mixed and ${rm H}$-mixed pseudo\u0000links. Finally, we extend the planar weighted resolution set to annular and\u0000toroidal pseudo knots, defining new invariants for classifying pseudo knots and\u0000links in the solid and in the thickened torus.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we calculate the annihilating polynomial of the colored Jones polynomial for the Hopf link and the Whitehead link and consider the link version of the AJ conjecture. We also give another annihilating polynomial of the colored Jones polynomial for those links.
本文计算了霍普夫链路和怀特海德链路的有色琼斯多项式的湮没多项式，并考虑了 AJ 猜想的链路转换。我们还给出了这些链路的有色琼斯多项式的另一个湮没多项式。
{"title":"On the Annihilating polynomial of the Colored Jones Polynomial for Some Links","authors":"Shun Sawabe","doi":"arxiv-2409.03802","DOIUrl":"https://doi.org/arxiv-2409.03802","url":null,"abstract":"In this paper, we calculate the annihilating polynomial of the colored Jones\u0000polynomial for the Hopf link and the Whitehead link and consider the link\u0000version of the AJ conjecture. We also give another annihilating polynomial of\u0000the colored Jones polynomial for those links.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joshua Broden, Malors Espinosa, Noah Nazareth, Niko Voth
We prove that all knots can be embedded into the Menger Sponge fractal. We prove that all Pretzel knots can be embedded into the Sierpinski Tetrahedron. Then we compare the number of iterations of each of these fractals needed to produce a given knot as a mean to compare the complexity of the two fractals.
{"title":"Knots Inside Fractals","authors":"Joshua Broden, Malors Espinosa, Noah Nazareth, Niko Voth","doi":"arxiv-2409.03639","DOIUrl":"https://doi.org/arxiv-2409.03639","url":null,"abstract":"We prove that all knots can be embedded into the Menger Sponge fractal. We\u0000prove that all Pretzel knots can be embedded into the Sierpinski Tetrahedron.\u0000Then we compare the number of iterations of each of these fractals needed to\u0000produce a given knot as a mean to compare the complexity of the two fractals.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive an analog of Mirzakhani's recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles $L geq 0$. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani's recursion as long as $L leq 2 sinh^{-1} 1$. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.
{"title":"Topological recursion for hyperbolic string field theory","authors":"Atakan Hilmi Fırat, Nico Valdes-Meller","doi":"arxiv-2409.02982","DOIUrl":"https://doi.org/arxiv-2409.02982","url":null,"abstract":"We derive an analog of Mirzakhani's recursion relation for hyperbolic string\u0000vertices and investigate its implications for closed string field theory.\u0000Central to our construction are systolic volumes: the Weil-Petersson volumes of\u0000regions in moduli spaces of Riemann surfaces whose elements have systoles $L\u0000geq 0$. These volumes can be shown to satisfy a recursion relation through a\u0000modification of Mirzakhani's recursion as long as $L leq 2 sinh^{-1} 1$.\u0000Applying the pants decomposition of Riemann surfaces to off-shell string\u0000amplitudes, we promote this recursion to hyperbolic string field theory and\u0000demonstrate the higher order vertices are determined by the cubic vertex\u0000iteratively for any background. Such structure implies the solutions of closed\u0000string field theory obey a quadratic integral equation. We illustrate the\u0000utility of our approach in an example of a stubbed scalar theory.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We characterise the set of fundamental groups for which there exist $n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is sufficiently large. In particular, for $n ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds.
{"title":"Generalised doubles and simple homotopy types of high dimensional manifolds","authors":"Csaba Nagy, John Nicholson, Mark Powell","doi":"arxiv-2409.03082","DOIUrl":"https://doi.org/arxiv-2409.03082","url":null,"abstract":"We characterise the set of fundamental groups for which there exist\u0000$n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple\u0000homotopy equivalent, when $n$ is sufficiently large. In particular, for $n ge\u000012$ even, we show that examples exist for any finitely presented group $G$ such\u0000that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands\u0000on previous work, where we constructed the first examples of even-dimensional\u0000manifolds that are homotopy equivalent but not simple homotopy equivalent. Our\u0000construction is based on doubles of thickenings, and a key ingredient of the\u0000proof is a formula for the Whitehead torsion of a homotopy equivalence between\u0000such manifolds.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Finitely many hypersurfaces are removed from unordered configuration spaces of $n$ points in $mathbb{C}$ to obtain a fibration over unordered configuration spaces of $n-1$ complex points. Fundamental groups of these restricted configuration spaces are computed in small dimensions.
{"title":"Restricted configuration spaces","authors":"Barbu Rudolf Berceanu","doi":"arxiv-2409.02586","DOIUrl":"https://doi.org/arxiv-2409.02586","url":null,"abstract":"Finitely many hypersurfaces are removed from unordered configuration spaces\u0000of $n$ points in $mathbb{C}$ to obtain a fibration over unordered\u0000configuration spaces of $n-1$ complex points. Fundamental groups of these\u0000restricted configuration spaces are computed in small dimensions.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod complements admit hyperbolic structures, but their geometric properties are yet to be well understood. In this paper, we provide upper and lower bounds for the volumes of all hyperbolic rod complements in terms of rod parameters, and show that these bounds may be loose in general. We introduce better and asymptotically sharp volume bounds for a family of rod complements. The bounds depend only on the lengths of the continued fractions formed from the rod parameters.
{"title":"Volume bounds for hyperbolic rod complements in the 3-torus","authors":"Norman Do, Connie On Yu Hui, Jessica S. Purcell","doi":"arxiv-2409.02357","DOIUrl":"https://doi.org/arxiv-2409.02357","url":null,"abstract":"The study of rod complements is motivated by rod packing structures in\u0000crystallography. We view them as complements of links comprised of Euclidean\u0000geodesics in the 3-torus. Recent work of the second author classifies when such\u0000rod complements admit hyperbolic structures, but their geometric properties are\u0000yet to be well understood. In this paper, we provide upper and lower bounds for\u0000the volumes of all hyperbolic rod complements in terms of rod parameters, and\u0000show that these bounds may be loose in general. We introduce better and\u0000asymptotically sharp volume bounds for a family of rod complements. The bounds\u0000depend only on the lengths of the continued fractions formed from the rod\u0000parameters.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Helena Bergold, Joachim Orthaber, Manfred Scheucher, Felix Schröder
Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the ErdH{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex~drawings.
{"title":"Holes in Convex and Simple Drawings","authors":"Helena Bergold, Joachim Orthaber, Manfred Scheucher, Felix Schröder","doi":"arxiv-2409.01723","DOIUrl":"https://doi.org/arxiv-2409.01723","url":null,"abstract":"Gons and holes in point sets have been extensively studied in the literature.\u0000For simple drawings of the complete graph a generalization of the\u0000ErdH{o}s--Szekeres theorem is known and empty triangles have been\u0000investigated. We introduce a notion of $k$-holes for simple drawings and study\u0000their existence with respect to the convexity hierarchy. We present a family of\u0000simple drawings without 4-holes and prove a generalization of Gerken's empty\u0000hexagon theorem for convex drawings. A crucial intermediate step will be the\u0000structural investigation of pseudolinear subdrawings in convex~drawings.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The construction of the Farey tessellation in the hyperbolic plane starts with a finitely generated group of symmetries of an ideal triangle, i.e. a triangle with all vertices on the boundary. It induces a remarkable fractal structure on the boundary of the hyperbolic plane, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a generalisation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towards a generalisation in the three-dimensional case. We introduce conformal bryophylla, a class of subsets of the boundary of the hyperbolic 3-space which possess fractal properties similar to the Farey tessellation. We classify all conformal bryophylla and study the properties of their limiting sets.
{"title":"Farey Bryophylla","authors":"Oleg Karpenkov, Anna Pratoussevitch","doi":"arxiv-2409.01621","DOIUrl":"https://doi.org/arxiv-2409.01621","url":null,"abstract":"The construction of the Farey tessellation in the hyperbolic plane starts\u0000with a finitely generated group of symmetries of an ideal triangle, i.e. a\u0000triangle with all vertices on the boundary. It induces a remarkable fractal\u0000structure on the boundary of the hyperbolic plane, encoding every element by\u0000the continued fraction related to the structure of the tessellation. The\u0000problem of finding a generalisation of this construction to the higher\u0000dimensional hyperbolic spaces has remained open for many years. In this paper\u0000we make the first steps towards a generalisation in the three-dimensional case.\u0000We introduce conformal bryophylla, a class of subsets of the boundary of the\u0000hyperbolic 3-space which possess fractal properties similar to the Farey\u0000tessellation. We classify all conformal bryophylla and study the properties of\u0000their limiting sets.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a surgery formula for the ordinary Seiberg-Witten invariants of smooth $4$-manifolds with $b_1 =1$. Our formula expresses the Seiberg-Witten invariants of the manifold after the surgery, in terms of the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space. This formula can be used to find exotic smooth structures on nonsimply connected $4$-manifolds, and gives a lower bound of the genus of an embedding surface in nonsimply connected $4$-manifolds. In forthcoming work, we will extend these results to give a surgery formula for the families Seiberg-Witten invariants.
{"title":"A surgery formula for Seiberg-Witten invariants","authors":"Haochen Qiu","doi":"arxiv-2409.02265","DOIUrl":"https://doi.org/arxiv-2409.02265","url":null,"abstract":"We prove a surgery formula for the ordinary Seiberg-Witten invariants of\u0000smooth $4$-manifolds with $b_1 =1$. Our formula expresses the Seiberg-Witten\u0000invariants of the manifold after the surgery, in terms of the original\u0000Seiberg-Witten moduli space cut down by a cohomology class in the configuration\u0000space. This formula can be used to find exotic smooth structures on nonsimply\u0000connected $4$-manifolds, and gives a lower bound of the genus of an embedding\u0000surface in nonsimply connected $4$-manifolds. In forthcoming work, we will\u0000extend these results to give a surgery formula for the families Seiberg-Witten\u0000invariants.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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